Any entire function of genus 1 which is positive on the positive real axis and which has only negative zeros decreases on some unbounded interval of the positive axis. The inverse of its reciprocal is shown to have an extension from that interval to a Pick-function in the upper half plane. A similar result holds for a class of entire functions of genus 2 and in particular the inverse function of Barnes’ double gamma function on a certain interval of the positive axis can be extended to a Pick-function. These results are proved using positive and negative definite kernels.